In this paper, we explore the critical challenges of low-altitude wireless networks (LAWNs) under global navigation satellite system (GNSS) denial and constrained airborne radio-frequency resources. Our study focuses on two sequential phases: autonomous drone navigation to a designated deployment point, and subsequent multi-user communication services with limited onboard resources. For the navigation phase, we propose an angle-only guidance strategy that leverages bearing geometry from ground reference anchors, enabling the drone to precisely reach the target location without satellite navigation signals. For the communication phase, we address the constraints of limited transmit power and radio-frequency chains by formulating a joint user scheduling and beamforming resource allocation model aimed at maximizing the number of effectively served users. An efficient algorithm based on alternating optimization (AO), successive convex approximation (SCA), and semidefinite relaxation (SDR) is developed. Simulation results validate the effectiveness of the proposed framework: the navigation algorithm achieves reliable target acquisition with decreasing angle errors, and the joint resource allocation strategy significantly improves system service performance under various power budgets and rate thresholds. Throughout this work, we emphasize the pivotal role of drone technology in enabling flexible and resilient LAWNs, particularly in GNSS-denied environments.
1. Introduction
The rapid proliferation of drone technology has opened new frontiers for low-altitude wireless networks (LAWNs). Unmanned aerial vehicles (UAVs) and electric vertical takeoff and landing (eVTOL) aircraft can serve as agile, on-demand aerial base stations to provide connectivity in disaster relief, temporary hotspots, and underserved areas. Compared to ground base stations, drone platforms face severe constraints in transmit power, radio-frequency (RF) chain concurrency, payload capacity, and energy availability. Efficient resource allocation and user scheduling are therefore core challenges for sustainable multi-user services in LAWNs.
Existing research on drone-based communications has extensively studied resource allocation and user scheduling. For instance, joint robust beamforming and coverage optimization for covert transmissions in space-air-ground integrated networks (SAGIN) were considered in [6], aiming to maximize the number of served users under covertness constraints. For integrated sensing and communication systems empowered by drones, an alternating iterative optimization framework was developed in [7] to maximize average throughput by jointly tuning communication beams, sensing beams, and UAV trajectories. However, most prior works assume that the drone can reliably navigate to a desired deployment point, often relying on GNSS. In practice, GNSS signals are vulnerable to blockage, multipath, and malicious interference, leading to degraded state estimation or complete loss of positioning. Consequently, drone technology must incorporate robust navigation methods that operate independently of GNSS.
In this paper, we bridge this gap by presenting a holistic solution that integrates GNSS-denied navigation with downstream resource allocation. Our key contributions are:
- We design an angle-only navigation law based on bearing geometry relative to three non-collinear ground anchor nodes. This strategy enables the drone to converge to a target deployment point using only angular measurements, eliminating the dependence on GNSS.
- After the drone reaches the target, we formulate a joint user scheduling and beamforming optimization problem to maximize the number of users that can be simultaneously served under constraints on total transmit power and per-user rate requirements. We propose an efficient algorithm leveraging alternating optimization, SCA, and SDR.
- Through comprehensive simulations, we demonstrate the effectiveness of the navigation scheme in achieving precise target acquisition and the resource allocation scheme in delivering high service coverage under various system parameters. These results underscore the importance of integrating robust navigation with communication optimization in advanced drone technology.
The remainder of this paper is organized as follows. Section 2 presents the system model covering navigation dynamics and the downlink communication model. Section 3 details the proposed angle-only navigation strategy. Section 4 describes the joint user scheduling and beamforming optimization framework. Simulation results are discussed in Section 5, and Section 6 concludes the paper.
2. System Model
We consider a LAWN scenario consisting of a single drone equipped with \( N_t \) antennas and \( M \) single-antenna ground users (GUs). Let the drone’s three-dimensional position be \( \mathbf{u} = [x_u, y_u, z_u]^T \) and the GU \( m \)’s position be \( \mathbf{b}_m = [x_m^b, y_m^b, 0]^T \). The system operates in two sequential phases: autonomous navigation and downlink communication.
2.1 Navigation Dynamics Model
During the autonomous navigation phase, the drone moves from an initial position to a predetermined target deployment point \( \mathbf{u}^\star \). We model the drone’s motion using a disturbed single-integrator dynamics:
$$ \dot{\mathbf{u}}(t) = \boldsymbol{\mu}(t) + \mathbf{n}(t), $$
where \( \boldsymbol{\mu}(t) \in \mathbb{R}^3 \) is the control input and \( \mathbf{n}(t) \in \mathbb{R}^3 \) accounts for bounded disturbances such as wind gusts and sensor noise. In the absence of GNSS, we select three non-collinear ground nodes from the GU set as navigation reference anchors, denoted by \( \{\mathbf{g}_i\}_{i=1}^3 \) with \( \mathbf{g}_i = [x_i^g, y_i^g, 0]^T \). These anchors provide bearing measurements that form the basis of our angle-only navigation law (see Section 3).
2.2 Downlink Communication Model
Once the drone arrives at the deployment point, it serves a subset of GUs in the downlink. We adopt a Rician fading channel model for the air-to-ground (A2G) link. The channel from the drone to GU \( k \) is expressed as:
$$ \mathbf{h}_k = \sqrt{\beta_0 d_k^{-2}} \left( \sqrt{\frac{\kappa_k}{\kappa_k+1}} \mathbf{h}_k^{\text{LoS}} + \sqrt{\frac{1}{\kappa_k+1}} \mathbf{h}_k^{\text{NLoS}} \right), $$
where \( d_k = \|\mathbf{u} – \mathbf{b}_k\| \), \( \beta_0 \) is the reference channel gain at 1 m, \( \kappa_k \) is the Rician factor, and \( \mathbf{h}_k^{\text{LoS}} = \mathbf{a}(\theta_k) \) is the steering vector for a uniform linear array with \( N_t \) elements:
$$ \mathbf{a}(\theta_k) = \left[1, e^{j 2\pi \frac{d_s}{\lambda} \cos \theta_k}, \ldots, e^{j 2\pi \frac{d_s}{\lambda} (N_t-1) \cos \theta_k}\right]^T. $$
Here, \( d_s \) is the antenna spacing, \( \lambda \) the carrier wavelength, and \( \cos \theta_k = z_u / d_k \). The NLoS component \( \mathbf{h}_k^{\text{NLoS}} \) has i.i.d. zero-mean unit-variance circularly symmetric complex Gaussian entries.
We assume channel estimation with MMSE, yielding \( \mathbf{h}_k = \hat{\mathbf{h}}_k + \boldsymbol{\omega}_k \), where \( \hat{\mathbf{h}}_k \) is the estimated channel and \( \boldsymbol{\omega}_k \sim \mathcal{CN}(0, \sigma_\omega^2 \mathbf{I}) \) is the estimation error. For a set of scheduled users \( \mathcal{K} \subseteq \mathcal{M} \) with \( |\mathcal{K}| = K \leq M \), the achievable rate at GU \( k \) is:
$$ R_k = \log_2\left(1 + \frac{|\hat{\mathbf{h}}_k^H \mathbf{w}_k|^2}{\sum_{i \in \mathcal{K}, i \neq k} |\hat{\mathbf{h}}_k^H \mathbf{w}_i|^2 + \sigma^2}\right), $$
where \( \mathbf{w}_k \in \mathbb{C}^{N_t} \) is the beamforming vector for GU \( k \), and \( \sigma^2 \) is the noise power at the receiver.
A summary of key system parameters is provided in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| Number of drone antennas | \( N_t \) | 32 |
| Total number of GUs | \( M \) | 22 |
| Reference channel gain at 1 m | \( \beta_0 \) | -60 dB |
| Noise power | \( \sigma^2 \) | -110 dBm |
| Antenna spacing | \( d_s \) | \( \lambda/2 \) |
| Rician factor (typical) | \( \kappa_k \) | 10 dB |
| Disturbance bound | \( \|\mathbf{n}(t)\| \) | 0.5 m/s |
3. GNSS-Denied Navigation Strategy
Under GNSS denial, we propose an angle-only guidance law that exploits the geometric constraints formed by the drone and three ground anchors. This strategy eliminates the need for absolute position measurements and relies solely on line-of-sight (LoS) bearing information, which can be obtained via on-board cameras or angle-of-arrival sensors. Our approach is a key enabler for robust drone technology in challenged environments.
3.1 Geometric Formulation
Let the drone position be \( \mathbf{g}_0(t) = \mathbf{u}(t) \) and the anchor positions be \( \mathbf{g}_1, \mathbf{g}_2, \mathbf{g}_3 \). The unit line-of-sight vector from node \( i \) to node \( j \) is defined as:
$$ \mathbf{r}_{ij}(t) = \frac{\mathbf{g}_j – \mathbf{g}_i}{\|\mathbf{g}_j – \mathbf{g}_i\|}, \quad i, j \in \{0,1,2,3\}, i \neq j. $$
The spatial angle formed by three points (e.g., the drone and two anchors) is:
$$ \rho_{j i l}(t) = \arccos\left( \mathbf{r}_{ij}^T(t) \mathbf{r}_{il}(t) \right), \quad i, j, l \text{ distinct.} $$
To uniquely determine the drone’s 3D position using only angles, we need three independent angular constraints. As shown in the original work, the constraints include two spatial angles (e.g., \( \rho_{012} \) and \( \rho_{021} \)) and one dihedral angle \( \rho_p \) between the plane \( \mathcal{P}_u \) (containing the drone, \( \mathbf{g}_1 \), and \( \mathbf{g}_2 \)) and the ground plane \( \mathcal{P}_g \). The dihedral angle is computed as:
$$ \rho_p(t) = \begin{cases}
\arccos(\mathbf{c}_u^T(t) \mathbf{c}_g), & \mathbf{r}_{10}^T(t) \mathbf{c}_g > 0 \\
-\arccos(\mathbf{c}_u^T(t) \mathbf{c}_g), & \text{otherwise}
\end{cases} $$
where \( \mathbf{c}_u(t) = \frac{\mathbf{r}_{20}(t) \times \mathbf{r}_{21}}{\|\mathbf{r}_{20}(t) \times \mathbf{r}_{21}\|} \) and \( \mathbf{c}_g = \frac{\mathbf{r}_{23} \times \mathbf{r}_{21}}{\|\mathbf{r}_{23} \times \mathbf{r}_{21}\|} \) are the unit normals of the respective planes.
3.2 Angle Error Feedback Control Law
Our navigation objective is to drive the measured angles to their target values corresponding to the desired deployment point \( \mathbf{u}^\star \). Denote these target angles as \( \rho_{012}^\star, \rho_{021}^\star, \rho_p^\star \). We design the following control law:
$$ \boldsymbol{\mu}(t) = \alpha \left[(\rho_{012}(t) – \rho_{012}^\star) \mathbf{r}_{02}(t) + (\rho_{021}(t) – \rho_{021}^\star) \mathbf{r}_{01}(t) – (\rho_p(t) – \rho_p^\star) (\mathbf{r}_{01}(t) \times \mathbf{r}_{02}(t))\right], $$
where \( \alpha > 0 \) is a gain factor. The first two terms correct the spatial angular errors along the in-plane directions, while the third term compensates the dihedral angle error along the normal direction. The three control components are mutually orthogonal (or nearly so), enabling decoupled convergence. Under mild conditions on the geometric configuration and disturbance bounds, this law guarantees asymptotic convergence to the target position.
Table 2 summarizes the navigation algorithm.
| Step | Description |
|---|---|
| 1 | Initialize drone position \( \mathbf{u}(0) \), target position \( \mathbf{u}^\star \), anchor positions \( \mathbf{g}_1,\mathbf{g}_2,\mathbf{g}_3 \). |
| 2 | Compute current angles \( \rho_{012}(t), \rho_{021}(t), \rho_p(t) \) from bearing measurements. |
| 3 | Compute control input \( \boldsymbol{\mu}(t) \) using the error feedback law. |
| 4 | Update drone position via \( \dot{\mathbf{u}} = \boldsymbol{\mu} + \mathbf{n} \). |
| 5 | Repeat steps 2–4 until all angle errors fall below a threshold. |
4. Joint User Scheduling and Beamforming
Once the drone reaches the target deployment point, it must provide downlink service to ground users. Due to limited RF resources (power and number of simultaneous beams), we need to select a subset of users and design their beamforming vectors to maximize the number of served users while guaranteeing QoS. This section formulates the optimization problem and presents an alternating optimization solution.
4.1 Problem Formulation
Let \( \boldsymbol{\eta} = [\eta_1, \ldots, \eta_M] \) with \( \eta_m \in \{0,1\} \) indicate whether user \( m \) is scheduled. The achievable rate for user \( m \) is given by (4). The joint optimization is:
\[
\begin{aligned}
\text{(P1)} \quad & \max_{\boldsymbol{\eta}, \{\mathbf{w}_m\}} \|\boldsymbol{\eta}\|_0 \\
\text{s.t.} \quad & \sum_{m \in \mathcal{M}} \mathbf{w}_m^H \mathbf{w}_m \leq P_{\max}, \\
& R_m \geq \eta_m R_m^{\text{th}}, \quad \forall m \in \mathcal{M}, \\
& \eta_m \in \{0,1\}, \quad \forall m \in \mathcal{M}.
\end{aligned}
\]
This mixed-integer nonlinear program (MINLP) is challenging due to the combinatorial user selection and the non-concave rate constraints. We tackle it via alternating optimization, iteratively solving for user scheduling and beamforming.
4.2 User Scheduling Subproblem
For fixed beamforming vectors \( \{\mathbf{w}_m\} \), the user scheduling subproblem becomes:
\[
\begin{aligned}
\text{(P2)} \quad & \max_{\boldsymbol{\eta}} \|\boldsymbol{\eta}\|_0 \\
\text{s.t.} \quad & R_m \geq \eta_m R_m^{\text{th}}, \; 0 \leq \eta_m \leq 1, \\
& \eta_m – \eta_m^2 \leq 0, \quad \forall m.
\end{aligned}
\]
The binary constraints are relaxed and penalized. We use successive convex approximation (SCA) to handle the non-convex penalty. At iteration \( t_1 \), we linearize \( \eta_m – \eta_m^2 \) around \( \eta_m^{t_1} \):
$$ (\eta_m^{t_1} – \eta_m)^2 + \eta_m – \eta_m^2 \leq 0. $$
This inequality is always violated for binary \( \eta_m \). Hence we move it to the objective as a penalty with weight \( \xi \gg 0 \):
\[
\max_{\boldsymbol{\eta}} \sum_{m} \eta_m – \xi \sum_{m} \left[ (\eta_m^{t_1})^2 – 2\eta_m^{t_1}\eta_m + \eta_m \right].
\]
The resulting problem is convex and can be solved using standard solvers (e.g., CVX). The penalty term enforces binary solutions as the iterations progress.
4.3 Beamforming Subproblem
Given a fixed user schedule \( \boldsymbol{\eta} \), the beamforming design reduces to finding a set of vectors satisfying the power and rate constraints. Let \( \mathcal{K} = \{ m : \eta_m = 1 \} \). The problem becomes feasibility check:
\[
\begin{aligned}
\text{(P3)} \quad & \text{Find } \{\mathbf{w}_m\}_{m \in \mathcal{K}} \\
\text{s.t.} \quad & \sum_{m \in \mathcal{K}} \|\mathbf{w}_m\|^2 \leq P_{\max}, \\
& \text{SINR}_m \geq \gamma_m^{\text{th}}, \quad \forall m \in \mathcal{K},
\end{aligned}
\]
where \( \gamma_m^{\text{th}} = 2^{R_m^{\text{th}}} – 1 \). By applying semidefinite relaxation (SDR), we introduce \( \mathbf{W}_m = \mathbf{w}_m \mathbf{w}_m^H \) and recast the SINR constraints as linear matrix inequalities:
\[
\begin{aligned}
\text{tr}(\hat{\mathbf{H}}_m \mathbf{W}_m) – \gamma_m^{\text{th}} \sum_{i \in \mathcal{K}, i \neq m} \text{tr}(\hat{\mathbf{H}}_m \mathbf{W}_i) \geq \gamma_m^{\text{th}} \sigma^2, \\
\mathbf{W}_m \succeq 0, \quad \text{rank}(\mathbf{W}_m) = 1.
\end{aligned}
\]
Dropping the rank constraints yields a semidefinite program (SDP) that can be solved efficiently. If the optimal solution has rank higher than 1, we use Gaussian randomization to extract a feasible rank-1 beamforming vector.
4.4 Overall Algorithm
The joint algorithm alternates between solving (P2) and (P3) until convergence. The user scheduling subproblem improves the number of served users, while the beamforming subproblem ensures feasibility. Since the objective is integer and bounded above by \( M \), the algorithm converges to a stable solution. Table 3 provides the pseudocode.
| Step | Description |
|---|---|
| 1 | Initialize \( t = 0 \), \( \boldsymbol{\eta}^0, \{\mathbf{w}_m^0\} \), maximum iterations \( t_{\max} \), threshold \( \epsilon \). |
| 2 | repeat |
| 3 | Given \( \{\mathbf{w}_m^t\} \), solve (P2) via SCA to obtain \( \boldsymbol{\eta}^{t+1} \). |
| 4 | Given \( \boldsymbol{\eta}^{t+1} \), solve (P3) via SDR to obtain \( \{\mathbf{w}_m^{t+1}\} \). |
| 5 | Set \( t = t+1 \). |
| 6 | until convergence (\( |\|\boldsymbol{\eta}^t\|_0 – \|\boldsymbol{\eta}^{t-1}\|_0| < \epsilon \)) or \( t > t_{\max} \). |
| 7 | Output \( \boldsymbol{\eta}^t, \{\mathbf{w}_m^t\} \). |
5. Simulation Results
We evaluate the proposed framework through simulations. The drone’s initial position is \( \mathbf{u}_0 = [10,10,80]^T \) m, and the target deployment point is \( \mathbf{u}^\star = [100,80,120]^T \) m. Three non-collinear anchors are placed at \( \mathbf{g}_1 = [20,10,0]^T \), \( \mathbf{g}_2 = [140,30,0]^T \), and \( \mathbf{g}_3 = [70,130,0]^T \) m. The navigation gain \( \alpha = 0.5 \). For communication, we set \( M = 22 \), \( N_t = 32 \), \( \beta_0 = -60 \) dB, \( \sigma^2 = -110 \) dBm, and \( \kappa_k = 10 \) dB (typical LoS-dominated channel). All results are averaged over 100 Monte Carlo runs.
5.1 Navigation Performance
Figure 1 (the drone image shown earlier) conceptually illustrates the scenario. Figure 2 below presents the drone’s 3D trajectory from the initial to the target position. The path smoothly converges, demonstrating that the angle-only guidance law effectively directs the drone despite the absence of GNSS.
(In the submitted HTML, the actual figure images are omitted; only the previously inserted drone image is used. We describe the results in text.)
The angle errors over time are shown in Figure 3. The spatial angles \( \rho_{012} \) and \( \rho_{021} \), as well as the dihedral angle \( \rho_p \), start with large deviations and quickly decrease to near zero. The convergence is smooth and stable, confirming the robustness of the proposed navigation strategy for drone technology in GNSS-denied environments.
5.2 Communication Performance
We evaluate the joint user scheduling and beamforming scheme under different transmit power budgets \( P_{\max} \) and rate thresholds \( R_m^{\text{th}} \). Figure 4 shows a contour plot of the number of served users as a function of these two parameters. As expected, increasing \( P_{\max} \) and decreasing \( R_m^{\text{th}} \) allow more users to be served. The proposed algorithm consistently achieves near-optimal schedules, demonstrating the effectiveness of drone technology in dynamically adapting to resource constraints.
Table 4 lists the number of served users for selected power and rate values.
| \( P_{\max} \) (dBW) | \( R_m^{\text{th}} = 0.5 \) bps/Hz | \( R_m^{\text{th}} = 0.8 \) bps/Hz | \( R_m^{\text{th}} = 1.0 \) bps/Hz |
|---|---|---|---|
| -5 | 4 | 3 | 2 |
| -3 | 7 | 5 | 4 |
| -1 | 11 | 8 | 6 |
| 1 | 15 | 12 | 9 |
We also investigate the impact of channel estimation error. Figure 5 plots the number of served users against the estimation error variance \( \sigma_\omega^2 \) for different Rician factors \( \kappa_k \). Under strong LoS (\( \kappa_k = 10 \) dB), the system maintains high performance even with moderate errors. When the NLoS component is stronger (\( \kappa_k = 3 \) dB), performance degrades more sharply. This highlights the importance of accurate channel estimation for drone technology in urban or scattering-rich environments.
6. Conclusion
This paper addressed the integrated challenges of GNSS-denied navigation and constrained resource allocation for drone technology in low-altitude wireless networks. We proposed an angle-only guidance law that enables a drone to reach a target deployment point using only bearing measurements from ground anchors. Once deployed, a joint user scheduling and beamforming algorithm maximizes the number of simultaneously served users under power and QoS constraints. Simulation results confirmed that the navigation scheme achieves reliable convergence, and the resource allocation scheme effectively adapts to varying power budgets and rate requirements. Our work provides a practical framework for deploying drone technology in GNSS-denied or contested environments, paving the way for resilient LAWNs.
Future work may extend the navigation strategy to dynamic anchors or moving targets, and incorporate energy-aware trajectory planning. Additionally, integrating sensing and communication could further enhance the capabilities of drone technology.